The Set of Prime Numbers

TL;DR

This paper presents a deterministic method to generate all prime numbers up to a given limit using specific selection rules based on perfect numbers, offering a new explicit relation for prime generation.

Contribution

It introduces a novel explicit relation and a deterministic framework for generating prime numbers based on perfect numbers and selection rules.

Findings

Prime distribution can be expressed deterministically.

A new explicit relation for prime generation is proposed.

Asymptotic behavior of primes is analyzed.

Abstract

In this work we show that the prime distribution is deterministic. Indeed the set of prime numbers P can be expressed in terms of two subsets of N using three specific selection rules, acting on two sets of prime candidates. The prime candidates are obtained in terms of the first perfect number. The asymptotic behaviour is also considered. We obtain for the first time an explicit relation for generating the full set P of prime numbers smaller than n or equal to n.